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Mathematical Modeling in Economics and Finance with Probability and Stochastic Processes Steven R. Dunbar September 14, 2016

Mathematical Modeling in Economics and Finance …sdunbar1/MathematicalFinance/Lessons/... · Mathematical Modeling in Economics and Finance with Probability and Stochastic Processes

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  • Mathematical Modelingin Economics and Finance

    with Probability and Stochastic Processes

    Steven R. Dunbar

    September 14, 2016

  • To my wife Charlene, who manages the finances so well.

  • Preface

    History of the Book

    This book started with one purpose and ended with a different purpose. In2002, a former student, one of the best I had taught, approached me with abook about mathematical finance in his hand. He wanted a reading courseabout the subject, because he was thinking about a career in the area. Iflipped through the book briefly, and saw that it was too advanced for areading course even with a very good student. I was aware that the topiccombined probability theory and partial differential equations, both interestsof mine. Instead of a reading course, I agreed to conduct a seminar onmathematical finance, if enough others had an interest. There were others,including undergraduates, graduate students in finance and economics andeven some faculty from business.

    I soon found that there were no books or introductions to the subjectsuitable for mathematics students at the upper undergraduate level. I beganto gather my seminar notes and organize them.

    After three years of the seminar, it grew into a popular course for senior-level students from mathematics, finance, actuarial science, computer scienceand engineering. The variety of students and their backgrounds refined thecontent of the course. The course focus was on combining finance concepts,especially derivative securities, with probability theory, difference equationsand differential equations to derive consequences, primarily about optionprices.

    In late 2008, security markets convulsed and the U. S. economy wentinto a deep recession. The causes were many, and are still debated, but

    iii

  • iv PREFACE

    one underlying cause was because mathematical methods had been appliedin financial situations where they did not apply [70]. At the same timefor different reasons, mathematical professional organizations urged a newemphasis on mathematical modeling. The course and the associated notesevolved in response, with an emphasis on uses and abuses of modeling.

    Additionally, a new paradigm in mathematical sciences combining model-ing, statistics, visualization, and computing with large data sets, sometimescalled big data, was maturing and becoming common. Big data is nowa source of employment for many mathematics majors. The topic of financeand economics is a leader in big data because of the existing large datasets and the measurable value in exploiting the data.

    The result is the current book combining modeling, probability theory,difference and differential equations focused on quantitative reasoning, dataanalysis, probability, and statistics for economics and finance. The bookuses all of these topics to investigate modern financial instruments that haveenormous economic influence, but are hidden from popular view becausemany people wrongly believe these topics are esoteric and difficult.

    Purpose of the Book

    The purpose is to provide a textbook for a capstone course focusing onmathematical modeling in economic and finance. There are already manyfine books about mathematical modeling in physical and biological sciences.This text is for an alternative course for students interested in economicsciences instead of the classical sciences. This book combines mathematicalmodeling, probability theory, difference and differential equations, numericalsolution and simulation and mathematical analysis in a single course for un-dergraduates in mathematical sciences. I hope the style is engaging enoughthat it can also be enjoyably read as an introduction by any individual in-terested in these topics.

    I understand that this introductory modeling approach makes seriousconcessions to completeness and depth, financial accuracy and mathematicalrigor. Phillip Protter is an expert on mathematical finance and in a reviewof an elementary text on mathematical finance [56] he makes the followingremarks:

    Mathematical finance . . . is a difficult subject, requiring abroad array of knowledge of subjects that are traditionally con-sidered hard to learn.

  • v

    The mathematics involved in the Black-Scholes paradigm ismeasure-theoretic probability theory, Brownian motion, stochas-tic processes including Markov processes and martingale theory,Itos stochastic calculus, stochastic differential equations, andpartial differential equations. Those prerequisites give one entryto the subject, which is why it is best taught to advanced Ph.D.students. One might expect an American undergraduate to knowcalculus-based probability theory and to have had some exposureto PDEs and perhaps, if one is lucky, an economics course ortwo, but not much more. Therefore, any attempt to teach such asubject to undergraduates is fraught with compromise . . .

    Perhaps it is the same with mathematical finance: it simplyis not (yet?) meant to be an undergraduate subject. In a waythat is too bad, because the subject is beautiful and powerful,and expertise in it is much needed in industry.

    Combining economic and financial modeling with probability, stochasticprocesses, and differential equations along with quantitative reasoning, anddata analysis with some simulation and computing provides an inviting entryinto deeper aspects of this beautiful and powerful subject.

    The goals of the book are:

    1. Understand the properties of stochastic processes such as sequencesof random variables, coin-flipping games, Brownian motion and thesolutions of stochastic differential equations as a means for modelingfinancial instruments for the management of risk.

    2. Use financial instruments for the management of risk as motivationsfor the detailed study of stochastic processes and solutions of stochasticdifferential equations.

    3. Introduce standard stochastic processes at the level of the classic refer-ences by Karlin and Taylor, and Feller. The book proves some mathe-matical statements at the level of elementary analysis, some more ad-vanced statements have heuristic motivation without proof, and someadvanced results are stated without proof.

    4. Emphasize the mathematical modeling process applied to a modernarea that is not based on physical science yet still leads to classical par-

  • vi PREFACE

    tial differential equations and numerical methods. The field of math-ematical finance is only 40 years old, uses leading-edge mathematicaland economic ideas, and has some controversial foundational hypothe-ses. Mathematical finance is also data-rich and even advanced resultsare testable in the market. Using ideas illustrated daily on newspaperfinancial pages, the book applies the full cycle of mathematical mod-eling and analysis in a non-trivial, but still accessible, way that haseconomic applications.

    5. The goal of the book is to reach a point where the students thoroughlyunderstand the derivation and modeling of financial instruments, ad-vanced financial models, advanced stochastic processes, partial differen-tial equations, and numerical methods at a level sufficient for beginninggraduate study in mathematics, finance, economics, actuarial science,and for entry-level positions in the sophisticated financial services in-dustry.

    The general area of stochastic processes and mathematical finance hasmany textbooks and monographs already. This book differs from them inthe following ways:

    1. Most books on stochastic processes have a variety of applications, whilethis book concentrates on financial instruments for the management ofrisk as motivations for the detailed study of mathematical modelingwith stochastic processes. The emphasis is on the modeling process,not the financial instruments.

    2. Most books on mathematical finance assume either prerequisite knowl-edge about financial instruments or sophisticated mathematical meth-ods, especially measure-based probability theory and martingale the-ory. This book serves as a introductory preparation for those texts.

    3. This book emphasizes the practice of mathematical modeling, includingpost-modeling analysis and criticism, making it suitable for a wideraudience.

    Intended Audience and Background

    This book is primarily for undergraduate students in mathematics, eco-nomics, finance, and actuarial science. Students in physical sciences, com-

  • vii

    puter science and engineering will also benefit from the book with its em-phasis on modeling and the uses and limits of modeling. Graduate studentsin economics, finance and business benefit from the non-measure theoreticbased introduction to mathematical finance and mathematical modeling.

    This book is for students after a course on calculus-based probabilitytheory. To understand the explanations and complete the exercises:

    1. The reader should be able to calculate joint probabilities of independentevents.

    2. The reader should be able to calculate binomial probabilities and nor-mal probabilities using direct calculation, tables and computer or cal-culator applications.

    3. The reader should be able to recognize common probability distribu-tions such as negative binomial probabilities and calculate probabilitiesfrom them.

    4. The reader should be able to calculate means and variances for commonprobability distributions.

    5. The reader should be familiar with common statistical concepts of pa-rameter point evaluations and confidence intervals and hypothesis test-ing

    6. The reader should have a familiarity with compound interest calcula-tions, both continuous compounding and periodic compounding.

    7. The reader should be able to be able to perform interest calculationsto find present values, future values, and simple annuities.

    The text also assumes general knowledge of linear algebra, especiallyabout solutions of linear non-homogeneous equations in linear spaces. Afamiliarity with solving difference equations, also called recurrence equationsand recursions, is helpful, but not essential. Where needed, the solution ofthe specific difference equations uses elementary methods without referenceto the general theory. Likewise, a familiarity with differential equations ishelpful but not essential since the text derives specific solutions when nec-essary, again without reference to the general theory. Naturally, a course indifferential equations will deepen understanding and provide another means

  • viii PREFACE

    for discussing mathematical modeling, since that is often the course wheremany students first encounter significant mathematical modeling of physicaland biological phenomena. Concepts from linear algebra also enter into thediscussions about Markov processes, but this text does not make the deeperconnections. Ideas from linear algebra pervade the operators and data struc-tures in the program scripts.

    Program Scripts

    An important feature of this book is the simulation scripts that accompanymost sections. The simulation scripts illustrate the concepts and theoremsof the section with numerical and graphical evidence. The scripts are partof the Rule of 3 teaching philosophy of presenting mathematical ideassymbolically, numerically and graphically.

    The programs are springboards for further experimentation, not finishedapps for casual everyday use. The scripts are minimal in size, in scope ofimplementation and with minimal output. The scripts are not complete,stand-alone, polished applications, rather they are proof-of-concept startingpoints. The reader should run the scripts to illustrate the ideas and providenumerical examples of the results in the section. The scripts provide a seedfor new scripts to increase the size and scope of the simulations. Increasingthe size can often demonstrate convergence or increase confidence in theresults of the section. Increasing the size can also demonstrate that althoughconvergence is mathematically guaranteed, sometimes the rate of convergenceis slow. The reader is also encouraged to change the output, to provide moreinformation, to add different graphical representations, and to investigaterates of convergence.

    The scripts are not specifically designed to be efficient, either in programlanguage implementation or in mathematical algorithm. Efficiency is notignored, but it is not the primary consideration in the construction of thescripts. Similarity of the program algorithm to the mathematical idea takesprecedence over efficiency. One noteworthy aspect of both the similarityand efficiency is that all the languages use vectorization along with othernotational simplifications such as recycling. Vectorized scripts look more likethe mathematical expressions found in the text, making the code easier tounderstand. Vectorized code often runs much faster than the correspondingcode containing loops.

    The scripts are not intended to be a tutorial on how to do mathemat-

  • ix

    ical programming in any of the languages. A description of the algorithmused in the scripts is in each section. The description is usually in full sen-tences rather than the more formal symbolic representation found in com-puter science pseudo-code. Given the description and some basic awarenessof programming ideas, the scripts provide multiple examples for study. Thescripts provide a starting point for investigating, testing and comparing lan-guage features from the documentation or from other sources. The scriptsuse good programming style whenever possible, but clarity, simplicity andsimilarity to the mathematics are primary considerations.

    Connections to MAA CUPM guidelines

    The nature of the text as an interdisciplinary capstone text intentionally ad-dresses each of the cognitive and content recommendations from the Mathe-matical Association of Americas Committee on the Undergraduate Curricu-lum for courses and programs in mathematical sciences.

    Cognitive Recommendations

    1. Students should develop effective thinking and communication skills.

    An emphasis in the text is on development, solution and subsequentcritical analysis of mathematical models in economics and finance. Ex-ercises in most sections ask students to write comparisons and criticalanalyses of the ideas, theories, and concepts.

    2. Students should learn to link applications and theory.

    The entire text is committed to linking methods and theories of proba-bility and stochastic processes and difference and differential equationsto modern applications in economics and finance. Each chapter has aspecific application of the methods to a model in economics and finance.

    3. Students should learn to use technological tools.

    Computing examples with modern scripting languages appear through-out and many exercises encourage further adaptation and experimen-tation with the scripts.

    4. Students should develop mathematical independence and experienceopen ended inquiry.

  • x PREFACE

    Many exercises encourage further experimentation, data exploration,and up-to-date comparisons with new or extended data.

    Content Recommendations

    1. Mathematical sciences major programs should include concepts andmethods from calculus and linear algebra.

    The text makes extensive use of calculus for continuous probability andthrough differential equations. The text extends some of the ideas ofcalculus to the domain of stochastic calculus. Linear algebra appearsthroughout, from the theory of solutions of linear non-homogeneousequations in linear spaces to operators and data structures in the pro-gram scripts

    2. Students majoring in the mathematical sciences should learn to read,understand, analyze, and produce proofs, at increasing depth as theyprogress through a major.

    Mathematical proofs are not emphasized, because rigorous methodsfor stochastic processes need measure-theoretic tools from probability.However, when elementary tools from analysis are familiar to students,then the text provides proofs or proof sketches. Derivation of the solu-tion of specific difference equations and differential equations appearsin detail but without reference to general solution methods.

    3. Mathematical sciences major programs should include concepts andmethods from data analysis, computing, and mathematical modeling.

    Mathematical modeling in economics and finance is the reason for thisbook. Collection and analysis of economic and financial data frompublic sources is emphasized throughout, and the exercises extend andrenew the data. Providing extensive simulation of concepts throughcomputing in modern scripting languages is provided throughout. Ex-ercises encourage the extension and adaptation of the scripts for moresimulation and data analysis.

    4. Mathematical sciences major programs should present key ideas andconcepts from a variety of perspectives to demonstrate the breadth ofmathematics.

  • xi

    As a text for a capstone course in mathematics, the text uses the multi-ple perspectives of mathematical modeling, ideas from calculus, proba-bility and statistics, difference equations, and differential equations, allfor the purposes of a deeper understanding of economics and finance.The book emphasizes mathematical modeling as a motivation for newmathematical ideas, and the application of known mathematical ideasas a framework for mathematical models. The book emphasizes dif-ference and differential equations to analyze stochastic processes. Theanalogies between fundamental ideas of calculus and ways to analyzestochastic processes is also emphasized.

    5. All students majoring in the mathematical sciences should experiencemathematics from the perspective of another discipline.

    The goal of the text focuses on mathematics modeling as a tool for un-derstanding economic and finance. Collection and analysis of economicand financial data from public sources using the mathematical tools isemphasized throughout,

    6. Mathematical sciences major programs should present key ideas fromcomplementary points of view: continuous and discrete; algebraic andgeometric; deterministic and stochastic; exact and approximate.

    The text consistently moves from discrete models in finance to contin-uous models in finance by developing discrete methods in probabilityinto continuous time stochastic process ideas. The text emphasizes thedifferences between exact mathematics and approximate models.

    7. Mathematical sciences major programs should require the study of atleast one mathematical area in depth, with a sequence of upper-levelcourses.

    The text is for a capstone course combining significant mathemati-cal modeling using probability theory, stochastic processes, differenceequations, differential equations to understand economic and financeat a level beyond the usual undergraduate approach to economic andfinance using only calculus ideas.

    8. Students majoring in the mathematical sciences should work, indepen-dently or in a small group, on a substantial mathematical project that

  • xii PREFACE

    involves techniques and concepts beyond the typical content of a singlecourse.

    Many of the exercises, especially those that extend the scripts or thatcall for more data and data analysis are suitable for projects done eitherindependently or in small groups.

    9. Mathematical sciences major programs should offer their students anorientation to careers in mathematics.

    Financial services, banking, insurance and risk, financial regulation,and data analysis combined with some knowledge of computing are allgrowth areas for careers for students from the mathematical sciences.This text is an introduction to all of those areas.

    Sources

    The Mathematical Association of America Committee on the UndergraduateProgram recommendations are from the 2015 CUPM Curriculum Guide toMajors in the Mathematical Sciences

    http://www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPMguide_print.pdfhttp://www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPMguide_print.pdf

  • Contents

    Preface iii

    1 Background Ideas 1

    1.1 Brief History of Mathematical Finance . . . . . . . . . . . . . 1

    1.2 Options and Derivatives . . . . . . . . . . . . . . . . . . . . . 11

    1.3 Speculation and Hedging . . . . . . . . . . . . . . . . . . . . . 19

    1.4 Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

    1.5 Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . 32

    1.6 Randomness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

    1.7 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . 57

    1.8 A Model of Collateralized Debt Obligations . . . . . . . . . . 66

    2 Binomial Option Pricing Models 77

    2.1 Single Period Binomial Models . . . . . . . . . . . . . . . . . . 77

    2.2 Multiperiod Binomial Tree Models . . . . . . . . . . . . . . . 88

    3 First Step Analysis for Stochastic Processes 101

    3.1 A Coin Tossing Experiment . . . . . . . . . . . . . . . . . . . 101

    3.2 Ruin Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . 116

    3.3 Duration of the Gamblers Ruin . . . . . . . . . . . . . . . . 133

    3.4 A Stochastic Process Model of Cash Management . . . . . . . 148

    4 Limit Theorems for Stochastic Processes 171

    4.1 Laws of Large Numbers . . . . . . . . . . . . . . . . . . . . . 171

    4.2 Moment Generating Functions . . . . . . . . . . . . . . . . . . 179

    4.3 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . 186

    4.4 The Absolute Excess of Heads over Tails . . . . . . . . . . . . 200

    xiii

  • xiv CONTENTS

    5 Brownian Motion 2135.1 Intuitive Introduction to Diffusions . . . . . . . . . . . . . . . 2135.2 The Definition of Brownian Motion and the Wiener Process . 2205.3 Approximation of Brownian Motion by Coin-Flipping Sums . . 2355.4 Transformations of the Wiener Process . . . . . . . . . . . . . 2445.5 Hitting Times and Ruin Probabilities . . . . . . . . . . . . . . 2545.6 Path Properties of Brownian Motion . . . . . . . . . . . . . . 2645.7 Quadratic Variation of the Wiener Process . . . . . . . . . . . 272

    6 Stochastic Calculus 2876.1 Stochastic Differential Equations and the Euler-Maruyama Method2876.2 Itos Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 3026.3 Properties of Geometric Brownian Motion . . . . . . . . . . . 3086.4 Models of Stock Market Prices . . . . . . . . . . . . . . . . . . 3206.5 Monte Carlo Simulation of Option Prices . . . . . . . . . . . . 335

    7 The Black-Scholes Model 3597.1 Derivation of the Black-Scholes Equation . . . . . . . . . . . . 3597.2 Solution of the Black-Scholes Equation . . . . . . . . . . . . . 3667.3 Put-Call Parity . . . . . . . . . . . . . . . . . . . . . . . . . . 3827.4 Implied Volatility . . . . . . . . . . . . . . . . . . . . . . . . . 3957.5 Sensitivity, Hedging and the Greeks . . . . . . . . . . . . . . 4057.6 Limitations of the Black-Scholes Model . . . . . . . . . . . . . 419

  • List of Figures

    1.1 This is not the market for options! . . . . . . . . . . . . . . . 13

    1.2 Intrinsic value of a call option. . . . . . . . . . . . . . . . . . . 14

    1.3 A schematic diagram of the cash flow in the gold arbitrageexample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

    1.4 The cycle of modeling. . . . . . . . . . . . . . . . . . . . . . . 34

    1.5 Schematic diagram of a pendulum. . . . . . . . . . . . . . . . 40

    1.6 The process of mathematical modeling according to Edwardsand Penney. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

    1.7 The process of mathematical modeling according to GlennLedder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

    1.8 Initial conditions for a coin flip, following Keller. . . . . . . . . 51

    1.9 Persi Diaconis mechanical coin flipper. . . . . . . . . . . . . . 52

    1.10 A tree of some stochastic processes, from most general at thetop to more specific at the end leaves. Stochastic processesstudied in this text are red. . . . . . . . . . . . . . . . . . . . 62

    1.11 Default probabilities as a function of both the tranche number0 to 100 and the base mortgage default probability 0.01 to 0.15. 71

    2.1 The single period binomial model. . . . . . . . . . . . . . . . . 79

    2.2 A binomial tree. . . . . . . . . . . . . . . . . . . . . . . . . . . 89

    2.3 Pricing a European call. . . . . . . . . . . . . . . . . . . . . . 91

    2.4 Pricing a European put. . . . . . . . . . . . . . . . . . . . . . 92

    3.1 Welcome to my casino! . . . . . . . . . . . . . . . . . . . . . . 102

    3.2 Where not to lose at the casino! . . . . . . . . . . . . . . . . . 103

    3.3 Several typical cycles in a model of the reserve requirement. . 150

    4.1 Block diagram of transform methods. . . . . . . . . . . . . . . 180

    xv

  • xvi LIST OF FIGURES

    4.2 Approximation of the binomial distribution with the normaldistribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

    4.3 The half-integer correction: In the figure the binomial proba-bility is 0.7487. The simple normal approximation is 0.6809,but with the half-integer correction the normal approximationis 0.7482. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

    4.4 Probability of the absolute excess of x heads or tails in 500tosses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

    5.1 Image of a possible random walk in phase line after an oddnumber of steps. . . . . . . . . . . . . . . . . . . . . . . . . . . 214

    5.2 Graph of the Dow-Jones Industrial Average from February 17,2015 to February 16, 2016 (blue line) and a random walk withnormal increments with the same initial value and variance(red line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

    5.3 A standardized density histogram of 1000 daily close-to-closereturns on the S & P 500 Index, from February 29, 2012 toMarch 1, 2012, up to February 21, 2016 to February 22, 2016. 225

    6.1 Simulation with the Euler-Maruyama method of a process de-fined by a stochastic differential equation. . . . . . . . . . . . 291

    6.2 The p.d.f. and c.d.f. for a lognormal random variable withm = 1, s = 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . 310

    6.3 The Wilshire 5000 Index from April 1, 2009 to December 31,2014 plotted in blue along with a Geometric Brownian Motionhaving the same mean, variance and starting value in red. . . 326

    6.4 The q-q plot for the normalized log-changes from the Wilshire5000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

    7.1 Value of the call option at maturity. . . . . . . . . . . . . . . . 375

    7.2 Value of the call option at various times. . . . . . . . . . . . . 375

    7.3 Value surface from the Black-Scholes formula. . . . . . . . . . 376

    7.4 Value of the put option at maturity. . . . . . . . . . . . . . . . 387

    7.5 Value of the call option at various times. . . . . . . . . . . . . 387

    7.6 Value surface from the Black-Scholes formula. . . . . . . . . . 388

  • LIST OF FIGURES xvii

    7.7 Schematic diagram of using Newtons Method to solve for im-plied volatility. The current call option value is plotted inblue. The value of the call option as a function of is plottedin black. The tangent line is plotted in red . . . . . . . . . . . 397

    7.8 Value of the call option at various times. . . . . . . . . . . . . 4067.9 Delta of the call option at various times. . . . . . . . . . . . . 4087.10 Illustration of the idea of volatility smile with implied volatility

    data from call options on the S & P 500 index (SPX). . . . . . 4217.11 A schematic diagram illustrating the idea of fat tails. . . . . . 422

  • xviii LIST OF FIGURES

  • Chapter 1

    Background Ideas

    1.1 Brief History of Mathematical Finance

    Section Starter Question

    Name as many financial instruments as you can, and name or describe themarket where you would buy them. Also describe the instrument as highrisk or low risk.

    Introduction

    One sometime hears that compound interest is the eighth wonder of theworld, or the stock market is just a big casino. These are colorful say-ings, maybe based in happy or bitter experience, but each focuses on only oneaspect of one financial instrument. The time value of money and uncertaintyare the central elements influencing the value of financial instruments. Con-sidering only the time aspect of finance, the tools of calculus and differentialequations are adequate. When considering only the uncertainty, the tools ofprobability theory illuminate the possible outcomes. Considering time anduncertainty together, we begin the study of advanced mathematical finance.

    Finance theory is the study of economic agents behavior allocatingfinancial resources and risks across alternative financial instruments over timein an uncertain environment. Familiar examples of financial instruments arebank accounts, loans, stocks, government bonds and corporate bonds. Manyless familiar examples abound. Economic agents are units who buy andsell financial resources in a market. Typical economic agents are individual

    1

  • 2 CHAPTER 1. BACKGROUND IDEAS

    investors, banks, businesses, mutual funds and hedge funds. Each agent hasmany choices of where to buy, sell, invest and consume assets. Each choicecomes with advantages and disadvantages. An agent distributes resourcesamong the many possible investments with a goal in mind, often maximumreturn or minimum risk.

    Advanced mathematical finance is the study of the more sophisticatedfinancial instruments called derivatives. A derivative is a financial agree-ment between two parties that depends on the future price or performanceof an underlying asset. Derivatives are so called not because they involve arate of change, but because their value is derived from the underlying asset.The underlying asset could be a stock, a bond, a currency, or a commod-ity. Derivatives have become one of the financial worlds most importantrisk-management tools. Finance is about shifting and distributing risk andderivatives are especially efficient for that purpose [50].

    Two common derivatives are futures and options. Futures trading, akey practice in modern finance, probably originated in seventeenth centuryJapan, but the idea goes as far back as ancient Greece. Options were a featureof the tulip mania in seventeenth century Holland. Modern derivativesdiffer from their predecessors in that they are usually specifically designed toobjectify and price financial risk.

    Derivatives come in many types. The most common examples are fu-tures, agreements to trade something at a set price at a given date; options,the right but not the obligation to buy or sell at a given price; forwards,like futures but traded directly between two parties instead of on exchanges;and swaps, exchanging flows of income from different investments to man-age different risk exposure. For example, one party in a deal may want thepotential of rising income from a loan with a floating interest rate, while theother might prefer the predictable payments ensured by a fixed interest rate.The name of this elementary swap is a plain vanilla swap. More complexswaps mix the performance of multiple income streams with varieties of risk[50]. Another more complex swap is a credit-default swap in which a sellerreceives a regular fee from the buyer in exchange for agreeing to cover lossesarising from defaults on the underlying loans. These swaps are somewhatlike insurance [50]. These more complex swaps are the source of controversysince many people believe that they are responsible for the collapse or near-collapse of several large financial firms in late 2008. As long as two parties arewilling to trade risks and can agree on a price they can craft a correspondingderivative from any financial instrument. Businesses use derivatives to shift

  • 1.1. BRIEF HISTORY OF MATHEMATICAL FINANCE 3

    risks to other firms, chiefly banks. About 95% of the worlds 500 biggest com-panies use derivatives. Markets called exchanges. are the usual place to buyand sell derivatives with standardized terms. Derivatives tailored for specificpurposes or risks are bought and sold over the counter from big banks.The over the counter market dwarfs the exchange trading. In November2009, the Bank for International Settlements put the face value of over thecounter derivatives at $604.6 trillion. Using face value is misleading, afterstripping out off-setting claims the residual value is $3.7 trillion, still a largefigure [66].

    Mathematical models in modern finance contain beautiful applicationsof differential equations and probability theory. Additionally, mathematicalmodels of modern financial instruments have had a direct and significantinfluence on finance practice.

    Early History

    The origins of much of the mathematics in financial models traces to LouisBacheliers 1900 dissertation on the theory of speculation in the Paris mar-kets. Completed at the Sorbonne in 1900, this work marks the twin birthsof both the continuous time mathematics of stochastic processes and thecontinuous time economics of option pricing. While analyzing option pric-ing, Bachelier provided two different derivations of the partial differentialequation for the probability density for the Wiener process or Brown-ian motion. In one of the derivations, he works out what is now calledthe Chapman-Kolmogorov convolution probability integral. Along the way,Bachelier derived the method of reflection to solve for the probability func-tion of a diffusion process with an absorbing barrier. Not a bad performancefor a thesis on which the first reader, Henri Poincare, gave less than a topmark! After Bachelier, option pricing theory laid dormant in the economicsliterature for over half a century until economists and mathematicians re-newed study of it in the late 1960s. Jarrow and Protter [31] speculate thatthis may have been because the Paris mathematical elite scorned economicsas an application of mathematics.

    Bacheliers work was 5 years before Albert Einsteins 1905 discovery ofthe same equations for his famous mathematical theory of Brownian motion.The editor of Annalen der Physik received Einsteins paper on Brownian mo-tion on May 11, 1905. The paper appeared later that year. Einstein proposeda model for the motion of small particles with diameters on the order of 0.001

  • 4 CHAPTER 1. BACKGROUND IDEAS

    mm suspended in a liquid. He predicted that the particles would undergomicroscopically observable and statistically predictable motion. The Englishbotanist Robert Brown had already reported such motion in 1827 while ob-serving pollen grains in water with a microscope. The physical motion is nowcalled Brownian motion in honor of Browns description.

    Einstein calculated a diffusion constant to govern the rate of motion ofsuspended particles. The paper was Einsteins justification of the molecu-lar and atomic nature of matter. Surprisingly, even in 1905 the scientificcommunity did not completely accept the atomic theory of matter. In 1908,the experimental physicist Jean-Baptiste Perrin conducted a series of experi-ments that empirically verified Einsteins theory. Perrin thereby determinedthe physical constant known as Avogadros number for which he won theNobel prize in 1926. Nevertheless, Einsteins theory was difficult to rigor-ously justify mathematically. In a series of papers from 1918 to 1923, themathematician Norbert Wiener constructed a mathematical model of Brow-nian motion. Wiener and others proved many surprising facts about hismathematical model of Brownian motion, research that continues today. Inrecognition of his work, his mathematical construction is often called theWiener process. [31]

    Growth of Mathematical Finance

    Modern mathematical finance theory begins in the 1960s. In 1965 the economistPaul Samuelson published two papers that argue that stock prices fluctuaterandomly [31]. One explained the Samuelson and Fama efficient marketshypothesis that in a well-functioning and informed capital market, asset-price dynamics are described by a model in which the best estimate of anassets future price is the current price (possibly adjusted for a fair expectedrate of return.) Under this hypothesis, attempts to use past price data orpublicly available forecasts about economic fundamentals to predict secu-rity prices cannot succeed. In the other paper with mathematician HenryMcKean, Samuelson shows that a good model for stock price movementsis Geometric Brownian Motion. Samuelson noted that Bacheliers modelfailed to ensure that stock prices would always be positive, whereas geomet-ric Brownian motion avoids this error [31].

    The most important development was the 1973 Black-Scholes model foroption pricing. The two economists Fischer Black and Myron Scholes (andsimultaneously, and somewhat independently, the economist Robert Merton)

  • 1.1. BRIEF HISTORY OF MATHEMATICAL FINANCE 5

    deduced an equation that provided the first strictly quantitative model forcalculating the prices of options. The key variable is the volatility of theunderlying asset. These equations standardized the pricing of derivativesin exclusively quantitative terms. The formal press release from the RoyalSwedish Academy of Sciences announcing the 1997 Nobel Prize in Economicsstates that they gave the honor for a new method to determine the value ofderivatives. Robert C. Merton and Myron S. Scholes have, in collaborationwith the late Fischer Black developed a pioneering formula for the valuationof stock options. Their methodology has paved the way for economic valua-tions in many areas. It has also generated new types of financial instrumentsand facilitated more efficient risk management in society.

    The Chicago Board Options Exchange (CBOE) began publicly tradingoptions in the United States in April 1973, a month before the official pub-lication of the Black-Scholes model. By 1975, traders on the CBOE wereusing the model to both price and hedge their options positions. In fact,Texas Instruments created a hand-held calculator specially programmed toproduce Black-Scholes option prices and hedge ratios.

    The basic insight underlying the Black-Scholes model is that a dynamicportfolio trading strategy in the stock can replicate the returns from an optionon that stock. This is hedging an option and it is the most important ideaunderlying the Black-Scholes-Merton approach. Much of the rest of the bookwill explain what that insight means and how to apply it to calculate optionvalues.

    The story of the development of the Black-Scholes-Merton option pricingmodel is that Black started working on this problem by himself in the late1960s. His idea was to apply the capital asset pricing model to value theoption in a continuous time setting. Using this idea, the option value satis-fies a partial differential equation. Black could not find the solution to theequation. He then teamed up with Myron Scholes who had been thinkingabout similar problems. Together, they solved the partial differential equa-tion using a combination of economic intuition and earlier pricing formulas.

    At this time, Myron Scholes was at MIT. So was Robert Merton, whowas applying his mathematical skills to problems in finance. Merton showedBlack and Scholes how to derive their differential equation differently. Mertonwas the first to call the solution the Black-Scholes option pricing formula.Mertons derivation used the continuous time construction of a perfectlyhedged portfolio involving the stock and the call option together with thenotion that no arbitrage opportunities exist. This is the approach we will

  • 6 CHAPTER 1. BACKGROUND IDEAS

    take. In the late 1970s and early 1980s mathematicians Harrison, Krepsand Pliska showed that a more abstract formulation of the solution as amathematical model called a martingale provides greater generality.

    By the 1980s, the adoption of finance theory models into practice wasnearly immediate. Additionally, the mathematical models used in financialpractice became as sophisticated as any found in academic financial research[48].

    Several explanations account for the different adoption rates of mathe-matical models into financial practice during the 1960s, 1970s and 1980s.Money and capital markets in the United States exhibited historically lowvolatility in the 1960s; the stock market rose steadily, interest rates were rel-atively stable, and exchange rates were fixed. Such simple markets providedlittle incentive for investors to adopt new financial technology. In sharp con-trast, the 1970s experienced several events that led to market change andincreasing volatility. The most important of these was the shift from fixedto floating currency exchange rates; the world oil price crisis resulting fromthe creation of the Middle East cartel; the decline of the United States stockmarket in 1973-1974 which was larger in real terms than any comparableperiod in the Great Depression; and double-digit inflation and interest ratesin the United States. In this environment, the old rules of thumb and sim-ple regression models were inadequate for making investment decisions andmanaging risk [48].

    During the 1970s, newly created derivative-security exchanges tradedlisted options on stocks, futures on major currencies and futures on U.S.Treasury bills and bonds. The success of these markets partly resulted fromincreased demand for managing risks in a volatile economic market. This suc-cess strongly affected the speed of adoption of quantitative financial models.For example, experienced traders in the over the counter market succeeded byusing heuristic rules for valuing options and judging risk exposure. Howeverthese rules of thumb were inadequate for trading in the fast-paced exchange-listed options market with its smaller price spreads, larger trading volumeand requirements for rapid trading decisions while monitoring prices in boththe stock and options markets. In contrast, mathematical models like theBlack-Scholes model were ideally suited for application in this new tradingenvironment [48].

    The growth in sophisticated mathematical models and their adoption intofinancial practice accelerated during the 1980s in parallel with the extraordi-nary growth in financial innovation. A wave of de-regulation in the financial

  • 1.1. BRIEF HISTORY OF MATHEMATICAL FINANCE 7

    sector was an important element driving innovation.Conceptual breakthroughs in finance theory in the 1980s were fewer and

    less fundamental than in the 1960s and 1970s, but the research resources de-voted to the development of mathematical models was considerably larger.Major developments in computing power, including the personal computerand increases in computer speed and memory enabled new financial mar-kets and expansions in the size of existing ones. These same technologiesmade the numerical solution of complex models possible. Faster computersalso speeded up the solution of existing models to allow virtually real-timecalculations of prices and hedge ratios.

    Ethical considerations

    According to M. Poovey [54], Enron developed new derivatives specifically totake advantage of de-regulation. Poovey says that derivatives remain largelyunregulated, for they are too large, too virtual, and too complex for industryoversight to police. In 1997-1998 the Financial Accounting Standards Board(an industry standards organization whose mission is to establish and im-prove standards of financial accounting) did try to rewrite the rules governingthe recording of derivatives, but ultimately they failed: in the 1999-2000 ses-sion of Congress, lobbyists for the accounting industry persuaded Congressto pass the Commodities Futures Modernization Act, which exempted orexcluded over the counter derivatives from regulation by the CommodityFutures Trading Commission, the federal agency that monitors the futuresexchanges. Current law requires only banks and other financial institutionsto reveal their derivatives positions. In contrast, Enron, originally an energyand commodities firm which collapsed in 2001 due to an accounting scandal,never registered as a financial institution and was never required to disclosethe extent of its derivatives trading.

    In 1995, the sector composed of finance, insurance, and real estate over-took the manufacturing sector in Americas gross domestic product. Bythe year 2000 this sector led manufacturing in profits. The Bank for In-ternational Settlements estimates that in 2001 the total value of derivativecontracts traded approached one hundred trillion dollars, which is approx-imately the value of the total global manufacturing production for the lastmillennium. In fact, one reason that derivatives trades have to be electronicinstead of involving exchanges of capital is that the amounts exceed the totalof the worlds physical currencies.

  • 8 CHAPTER 1. BACKGROUND IDEAS

    Prior to the 1970s, mathematical models had a limited influence on fi-nance practice. But since 1973 these models have become central in marketsaround the world. In the future, mathematical models are likely to have anindispensable role in the functioning of the global financial system includingregulatory and accounting activities.

    We need to seriously question the assumptions that make models ofderivatives work: the assumptions that the market follows probability modelsand the assumptions underneath the mathematical equations. But what ifmarkets are too complex for mathematical models? What if unprecedentedevents do occur, and when they do as we know they do what if theyaffect markets in ways that no mathematical model can predict? What ifthe regularity that all mathematical models assume ignores social and cul-tural variables that are not subject to mathematical analysis? Or what if themathematical models traders use to price futures actually influence the futurein ways that the models cannot predict and the analysts cannot govern?

    Any virtue can become a vice if taken to extreme, and just so with theapplication of mathematical models in finance practice. At times, the mathe-matics of the models becomes too interesting and we lose sight of the modelsultimate purpose. Futures and derivatives trading depends on the belief thatthe stock market behaves in a statistically predictable way; in other words,that probability distributions accurately describe the market. The mathe-matics is precise, but the models are not, being only approximations to thecomplex, real world. The practitioner should apply the models only tenta-tively, assessing their limitations carefully in each application. The beliefthat the market is statistically predictable drives the mathematical refine-ment, and this belief inspires derivative trading to escalate in volume everyyear.

    Financial events since late 2008 show that the concerns of the previ-ous paragraphs have occurred. In 2009, Congress and the Treasury Depart-ment considered new regulations on derivatives markets. Complex deriva-tives called credit default swaps appear to have used faulty assumptions thatdid not account for irrational and unprecedented events, as well as socialand cultural variables that encouraged unsustainable borrowing and debt.Extremely large positions in derivatives which failed to account for unlikelyevents caused bankruptcy for financial firms such as Lehman Brothers andthe collapse of insurance giants like AIG. The causes are complex, but criticsfix some of the blame on the complex mathematical models and the peoplewho created them. This blame results from distrust of that which is not un-

  • 1.1. BRIEF HISTORY OF MATHEMATICAL FINANCE 9

    derstood. Understanding the models and their limitations is a prerequisitefor creating a future which allows proper risk management.

    Sources

    This section is adapted from the articles Influence of mathematical mod-els in finance on practice: past, present and future by Robert C. Mertonin Mathematical Models in Finance edited by S. D. Howison, F. P. Kelly,and P. Wilmott, Chapman and Hall, 1995, (HF 332, M384 1995); In Honorof the Nobel Laureates Robert C. Merton and Myron S. Scholes: A Par-tial Differential Equation that Changed the World by Robert Jarrow in theJournal of Economic Perspectives, Volume 13, Number 4, Fall 1999, pages229-248; and R. Jarrow and P. Protter, A short history of stochastic inte-gration and mathematical finance the early years, 1880-1970, IMS LectureNotes, Volume 45, 2004, pages 75-91. Some additional ideas are drawn fromthe article Can Numbers Ensure Honesty? Unrealistic Expectations and theU.S. Accounting Scandal, by Mary Poovey, in the Notice of the AmericanMathematical Society, January 2003, pages 27-35.

    Key Concepts

    1. Finance theory is the study of economic agents behavior allocatingfinancial resources and risks across alternative financial instrumentsover time in an uncertain environment. Mathematics provides tools tomodel and analyze that behavior in allocation and time, taking intoaccount uncertainty.

    2. Louis Bacheliers 1900 math dissertation on the theory of speculationin the Paris markets marks the twin births of both the continuous timemathematics of stochastic processes and the continuous time economicsof option pricing.

    3. The most important theoretical development was the Black-Scholesmodel for option pricing published in 1973.

    4. The growth in sophisticated mathematical models and their adoptioninto financial practice accelerated during the 1980s in parallel withthe extraordinary growth in financial innovation. Major developmentsin computing power, including the personal computer and increases

  • 10 CHAPTER 1. BACKGROUND IDEAS

    in computer speed and memory enabled new financial markets andexpansions in the size of existing ones.

    Vocabulary

    1. Finance theory is the study of economic agents behavior allocatingfinancial resources and risks across alternative financial instrumentsover time in an uncertain environment.

    2. A derivative is a financial agreement between two parties that de-pends on the future price or performance of an underlying asset. Theunderlying asset could be a stock, a bond, a currency, or a commodity.

    3. Types of derivatives: Derivatives come in many types. The mostcommon examples are futures, agreements to trade something at aset price at a given date; options, the right but not the obligation tobuy or sell at a given price; forwards, like futures but traded directlybetween two parties instead of on exchanges; and swaps, exchangingone lot of obligations for another. Derivatives can be based on prettymuch anything as long as two parties are willing to trade risks and canagree on a price [66].

    Problems to Work for Understanding

    1. Write a short summary of the tulip mania in seventeenth centuryHolland.

    2. Write a short summary of the South Sea Island bubble in eighteenthcentury England.

    3. Pick a commodity and find current futures prices for that commodity.

    4. Pick a stock and find current options prices on that stock.

    Outside Readings and Links:

    1. History of the Black Scholes Equation Accessed Thu Jul 23, 2009 6:07AM

    2. Clip from The Trillion Dollar Bet Accessed Fri Jul 24, 2009 5:29 AM.

    http://www.optiontradingpedia.com/free_black_scholes_model.htmhttp://www.youtube.com/watch?v=xGfXyVtiB1E

  • 1.2. OPTIONS AND DERIVATIVES 11

    1.2 Options and Derivatives

    Section Starter Question

    Suppose your rich neighbor offered an agreement to you today to sell his clas-sic Jaguar sports-car to you (and only you) a year from today at a reasonableprice agreed upon today. (You and your neighbor will exchange cash and cara year from today.) What would be the advantages and disadvantages to youof such an agreement? Would that agreement be valuable? How would youdetermine how valuable that agreement is?

    Definitions

    A call option is the right to buy an asset at an established price at a certaintime. A put option is the right to sell an asset at an established price at acertain time. Another slightly simpler financial instrument is a future whichis a contract to buy or sell an asset at an established price at a certain time.

    More fully, a call option is an agreement or contract by which at a defi-nite time in the future, known as the expiry date, the holder of the optionmay purchase from the option writer an asset known as the underlyingasset for a definite amount known as the exercise price or strike price.A put option is an agreement or contract by which at a definite time inthe future, known as the expiry date, the holder of the option may sell tothe option writer an asset known as the underlying asset for a definiteamount known as the exercise price or strike price. The holder of a Eu-ropean option may only exercise it at the end of its life on the expiry date.The holder of an American option may exercise it at any time during itslife up to the expiry date. For comparison, in a futures contract the writermust buy (or sell) the asset to the holder at the agreed price at the prescribedtime. The underlying assets commonly traded on options exchanges includestocks, foreign currencies, and stock indices. For futures, in addition to thesekinds of assets the common assets are commodities such as minerals andagricultural products. In this text we will usually refer to options based onstocks, since stock options are easily described, commonly traded and pricesare easily found.

    Jarrow and Protter [31, page 7] tell a story about the origin of the namesEuropean options and American options. While writing his important 1965article on modeling stock price movements as a geometric Brownian motion,

  • 12 CHAPTER 1. BACKGROUND IDEAS

    Paul Samuelson went to Wall Street to discuss options with financial profes-sionals. Samuelsons Wall Street contact informed him that there were twokinds of options, one more complex that could be exercised at any time, theother more simple that could be exercised only at the maturity date. Thecontact said that only the more sophisticated European mind (as opposedto the American mind) could understand the former more complex option.In response, when Samuelson wrote his paper, he used these prefixes andreversed the ordering! Now in a further play on words, financial marketsoffer many more kinds of options with geographic labels but no relation tothat place name. For example; two common types are Asian options andBermuda options.

    The Markets for Options

    In the United States, some exchanges trading options are the Chicago BoardOptions Exchange (CBOE), the American Stock Exchange (AMEX), and theNew York Stock Exchange (NYSE) among others. Not all options trade onexchanges. Over-the-counter options markets where financial institutions andcorporations trade directly with each other are increasingly popular. Tradingis particularly active in options on foreign exchange and interest rates. Themain advantage of an over-the-counter option is that a financial institutioncan tailor it to meet the needs of a particular client. For example, the strikeprice and maturity do not have to correspond to the set standards of theexchanges. The parties to the option can incorporate other nonstandardfeatures into the option. A disadvantage of over-the-counter options is thatthe terms of the contract need not be open to inspection by others and thecontract may be so different from standard derivatives that it is hard toevaluate in terms of risk and value.

    A European put option allows the holder to sell the asset on a certaindate for a prescribed amount. The put option writer is obligated to buy theasset from the option holder. If the underlying asset price goes below thestrike price, the holder makes a profit because the holder can buy the assetat the current low price and sell it at the agreed higher price instead of thecurrent price. If the underlying asset price goes above the strike price, theholder exercises the right not to sell. The put option has payoff propertiesthat are the opposite to those of a call. The holder of a call option wants theasset price to rise, the higher the asset price, the higher the immediate profit.The holder of a put option wants the asset price to fall as low as possible.

  • 1.2. OPTIONS AND DERIVATIVES 13

    Figure 1.1: This is not the market for options!

    The further below the strike price, the more valuable is the put option.

    The expiry date specifies the month in which the European option ends.The precise expiration date of exchange traded options is 10:59 PM CentralTime on the Saturday immediately following the third Friday of the expi-ration month. The last day on which options trade is the third Friday ofthe expiration month. Exchange traded options are typically offered withlifetimes of 1, 2, 3, and 6 months.

    Another item used to describe an option is the strike price, the priceat which the asset can be bought or sold. For exchange traded options onstocks, the exchange typically chooses strike prices spaced $2.50, $5, or $10apart. The usual rule followed by exchanges is to use a $2.50 spacing if thestock price is below $25, $5 spacing when it is between $25 and $200, and$10 spacing when it is above $200. For example, if Corporation XYZ hasa current stock price of $12.25, options traded on it may have strike pricesof $10, $12.50, $15, $17.50 and $20. A stock trading at $99.88 may haveoptions traded at the strike prices of $90, $95, $100, $105, $110 and $115.

    Options can be in the money, at the money or out of the money.An in-the-money option would lead to a positive cash flow to the holder if itwere exercised immediately. Similarly, an at-the-money option would lead tozero cash flow if exercised immediately, and an out-of-the-money would leadto negative cash flow if it were exercised immediately. If S is the stock priceand K is the strike price, a call option is in the money when S > K, at the

  • 14 CHAPTER 1. BACKGROUND IDEAS

    Option

    IntinsicValue

    Stock PriceK

    Figure 1.2: Intrinsic value of a call option.

    money when S = K and out of the money when S < K. Clearly, an optionwill be exercised only when it is in the money.

    Characteristics of Options

    The intrinsic value of an option is the maximum of zero and the value itwould have if exercised immediately. For a call option, the intrinsic valueis therefore max(S K, 0). Often it might be optimal for the holder of anAmerican option to wait rather than exercise immediately. The option is thensaid to have time value. Note that the intrinsic value does not consider thetransaction costs or fees associated with buying or selling an asset.

    The word may in the description of options, and the name optionitself implies that for the holder of the option or contract, the contract is aright, and not an obligation. The other party of the contract, known as thewriter does have a potential obligation, since the writer must sell (or buy)the asset if the holder chooses to buy (or sell) it. Since the writer confers onthe holder a right with no obligation an option has some value. The holdermust pay for the right at the time of opening the contract. Conversely, thewriter of the option must be compensated for the obligation taken on. Ourmain goal is to answer the following questions:

    How much should one pay for that right? That is, what is thevalue of an option? How does that value vary in time? How doesthat value depend on the underlying asset?

    Note that the value of the option contract depends essentially on the char-acteristics of the underlying asset. If the asset has relatively large variations

  • 1.2. OPTIONS AND DERIVATIVES 15

    Increase in European European American AmericanVariable Call Put Call PutStock Price Increase Decrease Increase DecreaseStrike Price Decrease Increase Decrease IncreaseTime to Expiration ? ? Increase IncreaseVolatility Increase Increase Increase IncreaseRisk-free Rate Increase Decrease Increase DecreaseDividends Decrease Increase Decrease Increase

    Table 1.1: Effect on price of increases in the variables influencing optionprices.

    in price, then we might believe that the option contract would be relativelyhigh-priced since with some probability the option will be in the money. Theoption contract value is derived from the asset price, and so we call it aderivative.

    Six factors affect the price of a stock option:

    the current stock price S;

    the strike price K;

    the time to expiration T t where T is the expiration time and t is thecurrent time;

    the volatility of the stock price;

    the risk-free interest rate; and

    the dividends expected during the life of the option.

    Consider what happens to option prices when one of these factors changeswhile all the others remain fixed. Table 1.1 summarizes the results. Thechanges regarding the stock price, the strike price, the time to expirationand the volatility are easy to explain; the other variables are less importantfor our considerations.

    Upon exercising it at some time in the future, the payoff from a call optionwill be the amount by which the stock price exceeds the strike price. Calloptions therefore become more valuable as the stock price increases and lessvaluable as the strike price increases. For a put option, the payoff on exercise

  • 16 CHAPTER 1. BACKGROUND IDEAS

    is the amount by which the strike price exceeds the stock price. Put optionstherefore behave in the opposite way to call options. Put options become lessvaluable as stock price increases and more valuable as strike price increases.

    Consider next the effect of the expiration date. Both put and call Amer-ican options become more valuable as the time to expiration increases. Theowner of a long-life option has all the exercise options open to the short-life option and more. The long-life option must therefore, be worth atleast as much as the short-life option. European put and call options do notnecessarily become more valuable as the time to expiration increases. Theowner of a long-life European option can only exercise at the maturity of theoption.

    Roughly speaking, the volatility of a stock price is a measure of howmuch future stock price movements may vary relative to the current price.As volatility increases, the chance that the stock price will either increase ordecrease greatly relative to the present price also increases. For the owner ofa stock, these two outcomes tend to offset each other. However, this is notso for the owner of a put or call option. The owner of a call benefits fromprice increases, but has limited downside risk in the event of price decreasesince the most that he or she can lose is the price of the option. Similarly,the owner of a put benefits from price decreases but has limited upside riskin the event of price increases. The values of puts and calls therefore increaseas volatility increases.

    The reader will observe that the language about option prices in thissection has been qualitative and imprecise:

    an option is a contract to buy or sell an asset at an established pricewithout specifying how the price is obtained;

    . . . the option contract would be relatively high-priced . . . ;

    Call options therefore become more valuable as the stock price in-creases . . . without specifiying the rate of change; and

    As volatility increases, the chance that the stock price will either in-crease or decrease greatly . . . increases.

    The goal in following sections is to develop a mathematical model whichgives quantitative and precise statements about options prices and to judgethe validity and reliability of the model.

  • 1.2. OPTIONS AND DERIVATIVES 17

    Sources

    The ideas in this section are adapted from Options, Futures and other Deriva-tive Securities by J. C. Hull, Prentice-Hall, Englewood Cliffs, New Jersey,1993 and The Mathematics of Financial Derivatives by P. Wilmott, S. How-ison, J. Dewynne, Cambridge University Press, 195, Section 1.4, What areoptions for?, Page 13 and R. Jarrow and P. Protter, A short history ofstochastic integration and mathematical finance the early years, 18801970,IMS Lecture Notes, Volume 45, 2004, pages 7591.

    Key Concepts

    1. A call option is the right to buy an asset at an established price at acertain time.

    2. A put option is the right to sell an asset at an established price at acertain time.

    3. A European option may only be exercised at the end of its life on theexpiry date, an American option may be exercised at any time duringits life up to the expiry date.

    4. Six factors affect the price of a stock option:

    the current stock price S; the strike price K; the time to expiration T t where T is the expiration time and t

    is the current time;

    the volatility of the stock price ; the risk-free interest rate r; and the dividends expected during the life of the option.

    Vocabulary

    1. A call option is the right to buy an asset at an established price at acertain time.

    2. A put option is the right to sell an asset at an established price at acertain time.

  • 18 CHAPTER 1. BACKGROUND IDEAS

    3. A future is a contract to buy (or sell) an asset at an established priceat a certain time.

    4. Volatility is a measure of the variability and therefore the risk of aprice, usually the price of a security.

    Problems to Work for Understanding

    1. (a) Find and write the definition of a future, also called a futurescontract. Graph the intrinsic value of a futures contract at itscontract date, or expiration date, as in 1.2.

    (b) Explain why holding a call option and writing a put option withthe same strike price K on the same asset is the same as having afutures contract on the asset with strike price K. Draw a graph ofthe value of the option combination and the value of the futurescontract on the same axes.

    2. Puts and calls are not the only option contracts available, just the mostfundamental and the simplest. Puts and calls eliminate the risk of upor down price movements in the underlying asset. Some other optioncontracts designed to eliminate other risks are combinations of putsand calls.

    (a) Draw the graph of the value of the option contract composed ofholding a put option with strike price K1 and holding a call optionwith strike price K2 where K1 < K2. (Assume both the put andthe call have the same expiration date.) The holder profits onlyif the underlier moves dramatically in either direction. This isknown as a long strangle.

    (b) Draw the graph of the value of an option contract composed ofholding a put option with strike price K and holding a call optionwith the same strike price K. (Assume both the put and the callhave the same expiration date.) This is called an long straddle,and also called a bull straddle.

    (c) Draw the graph of the value of an option contract composed ofholding one call option with strike price K1 and the simultaneous

  • 1.3. SPECULATION AND HEDGING 19

    writing of a call option with strike price K2 with K1 < K2. (As-sume both the options have the same expiration date.) This isknown as a bull call spread.

    (d) Draw the graph of the value of an option contract created by si-multaneously holding one call option with strike price K1, holdinganother call option with strike price K2 where K1 < K2, and writ-ing two call options at strike price (K1 +K2)/2. This is known asa butterfly spread.

    (e) Draw the graph of the value of an option contract created byholding one put option with strike price K and holding two calloptions on the same underlying security, strike price, and maturitydate. This is known as a triple option or strap

    Outside Readings and Links:

    1. What are stock options? An explanation from youtube.com

    1.3 Speculation and Hedging

    Section Starter Question

    Discuss examples of speculation in your experience. (Example: think ofscalping tickets.) A hedge is a transaction or investment that is taken outspecifically to reduce or cancel out risk. Discuss examples of hedges in yourexperience.

    Definitions

    Options have two primary uses, speculation and hedging. Speculationis to assume a financial risk in anticipation of a gain, especially to buy orsell to profit from market fluctuations. The market fluctuations are randomfinancial variations with a known (or assumed) probability distribution.

    Risk and Uncertainty

    Risk, first articulated by the economist F. Knight in 1921, is a variabilitythat you can put a price on. That is, risk is random financial variation

    http://www.youtube.com/watch?v=EH9RjItbR00

  • 20 CHAPTER 1. BACKGROUND IDEAS

    that has a known (or assumed) probability distribution. In poker, say thatyoull win a poker hand unless your opponent draws to an inside straight,a particular kind of card draw from the deck. It is not necessary to knowwhat this poker play means. However what is important to know is that thisparticular kind of draw has a probability of exactly 1/11. A poker playercan calculate the 1/11 with simple rules of probability theory. Your betis risk, you gain or lose of your bet with a known probability. It may beunpleasant to lose the bet, but at least you can account for it in advancewith a probability, [65].

    Uncertainty is chance variability due to unknown and unmeasured fac-tors. You might have some awareness (or not) of the variability out there.You may have no idea of how many such factors exist, or when any one maystrike, or how big the effects will be. Uncertainty is the unknown unknowns[65].

    Risk sparks a free-market economy with the impulse to make a gain.Uncertainty halts an economy with fear.

    Example: Speculation on a stock with calls

    An investor who believes that a particular stock, say XYZ, is going to risemay purchase some shares in the company. If she is correct, she makesmoney, if she is wrong she loses money. The investor is speculating. Supposethe price of the stock goes from $2.50 to $2.70, then the investor makes $0.20on each $2.50 investment, or a gain of 8%. If the price falls to $2.30, thenthe investor loses $0.20 on each $2.50 share, for a loss of 8%. These are bothstandard calculations.

    Alternatively, suppose the investor thinks that the share price is goingto rise within the next couple of months, and that the investor buys a calloption with exercise price of $2.50 and expiry date in three months.

    Now assume that it costs $0.10 to purchase a European call option onstock XYZ with expiration date in three months and strike price $2.50. Thatmeans in three months time, the investor could, if the investor chooses to,purchase a share of XYZ at price $2.50 per share no matter what the currentprice of XYZ stock is! Note that the price of $0.10 for this option may notbe an proper price for the option, but we use $0.10 simply because it is easyto calculate with. However, 3-month option prices are often about 5% of thestock price, so $0.10 is reasonable. In three months time if the XYZ stockprice is $2.70, then the holder of the option may purchase the stock for $2.50.

  • 1.3. SPECULATION AND HEDGING 21

    This action is called exercising the option. It yields an immediate profit of$0.20. That is, the option holder can buy the share for $2.50 and immediatelysell it in the market for $2.70. On the other hand if in three months time,the XYZ share price is only $2.30, then it would not be sensible to exercisethe option. The holder lets the option expire. Now observe carefully: Bypurchasing an option for $0.10, the holder can derive a net profit of $0.10($0.20 revenue less $0.10 cost) or a loss of $0.10 (no revenue less $0.10 cost.)The profit or loss is magnified to 100% with the same probability of change.Investors usually buy options in quantities of hundreds, thousands, even tensof thousands so the absolute dollar amounts can be large. Compared withstocks, options offer a great deal of leverage, that is, large relative changes invalue for the same investment. Options expose a portfolio to a large amountof risk cheaply. Sometimes a large degree of risk is desirable. This is the useof options and derivatives for speculation.

    Example: Speculation on a stock with calls

    Consider the profit and loss of a investor who buys 100 call options on XYZstock with a strike price of $140. Suppose the current stock price is $138, theexpiration date of the option is two months, and the option price is $5. Sincethe options are European, the investor can exercise only on the expirationdate. If the stock price on this date is less than $140, the investor will choosenot to exercise the option since buying a stock at $140 that has a marketvalue less than $140 is not sensible. In these circumstances the investor losesthe whole of the initial investment of $500. If the stock price is above $140on the expiration date, the holder will exercise the options. Suppose forexample,the stock price is $155. By exercising the options, the investor isable to buy 100 shares for $140 per share. By selling the shares immediately,the investor makes a gain of $15 per share, or $1500 ignoring transactioncosts. Taking the initial cost of the option into account, the net profit to theinvestor is $10 per option, or $1000 on an initial investment of $500. Notethat this calculation ignores any time value of money.

    Example: Speculation on a stock with puts

    Consider an investor who buys 100 European put options on XYZ with astrike price of $90. Suppose the current stock price is $86, the expirationdate of the option is in 3 months and the option price is $7. Since the

  • 22 CHAPTER 1. BACKGROUND IDEAS

    options are European, the holder will exercise only if the stock price is below$90 at the expiration date. Suppose the stock price is $65 on this date. Theinvestor can buy 100 shares for $65 per share, and under the terms of theput option, sell the same stock for $90 to realize a gain of $25 per share,or $2500. Again, this simple example ignores transaction costs. Taking theinitial cost of the option into account, the investors net profit is $18 peroption, or $1800. This is a profit of 257% even though the stock has onlychanged price $25 from an initial of $90, or 28%. Of course, if the final priceis above $90, the put option expires worthless, and the investor loses $7 peroption, or $700.

    Example: Hedging with calls on foreign exchange rates

    Suppose that a U.S. company knows that it is due to pay 1 million pounds toa British supplier in 90 days. The company has significant foreign exchangerisk. The cost in U.S. dollars of making the payment depends on the exchangerate in 90 days. The company instead can buy a call option contract toacquire 1 million pounds at a certain exchange rate, say 1.7 in 90 days. If theactual exchange rate in 90 days proves to be above 1.7, the company exercisesthe option and buys the British pounds it requires for $1,700,000. If theactual exchange rate proves to be below 1.7, the company buys the poundsin the market in the usual way. This option strategy allows the companyto insure itself against adverse exchange rate increases but still benefit fromfavorable decreases. Of course this insurance comes at the relatively smallcost of buying the option on the foreign exchange rate.

    Example: Hedging with a portfolio with puts and calls

    Since the value of a call option rises when an asset price rises, what happensto the value of a portfolio containing both shares of stock of XYZ and anegative position in call options on XYZ stock? If the stock price is rising,the call option value will also rise, the negative position in calls will becomegreater, and the net portfolio should remain approximately constant if thepositions are in the right ratio. If the stock price is falling then the calloption value price is also falling. The negative position in calls will becomesmaller. If held in the proper amounts, the total value of the portfolio shouldremain constant! The risk (or more precisely, the variation) in the portfoliois reduced! The reduction of risk by taking advantage of such correlations

  • 1.3. SPECULATION AND HEDGING 23

    is called hedging. Used carefully, options are an indispensable tool of riskmanagement.

    Consider a stock currently selling at $100 and having a standard deviationin its price fluctuations of $10, for a proportion variation of 10%. We canuse the Black-Scholes formula derived later to show that a call option with astrike price of $100 and a time to expiration of one year would sell for $11.84.A 1 percent rise in the stock from $100 to $101 would drive the option priceto $12.73. Consider the total effects in Table 1.2.

    Suppose a trader has an original portfolio comprised of 8944 shares ofstock selling at $100 per share. (The unusual number of 8944 shares comesfrom the Black-Scholes formula as a hedge ratio.) Assume also that a tradershort sells call options on 10,000 shares at the current price of $11.84. Thatis, the short seller borrows the options from another trader and thereforemust later return the options at the option price at the return time. Theobligation to return the borrowed options creates a negative position in theoption value. The transaction is called short selling because the trader sells agood he or she does not actually own and must later pay it back. In Table 1.2this debt or short position in the option is indicated by a minus sign. Theentire portfolio of shares and options has a net value of $776,000.

    Now consider the effect of a 1 percent change in the price of the stock. Ifthe stock increases 1 percent,the shares will be worth $903,344. The optionprice will increase from $11.84 to $12.73. But since the portfolio also involvesa short position in 10,000 options, this creates a loss of $8,900. This is theadditional value of what the borrowed options are now worth, so the borrowermust additionally this amount back! Taking these two effects into account,the value of the portfolio will be $776,044. This is nearly the same as theoriginal value. The slight discrepancy of $44 is rounding error due to the factthat the number of stock shares calculated from the hedge ratio is roundedto an integer number of shares for simplicity in the example, and the changein option value is rounded to the nearest penny, also for simplicity. In actualpractice, financial institutions take great care to avoid round-off differences.

    On the other hand of the stock price falls by 1 percent, there will be aloss in the stock of $8944. The price on this option will fall from $11.84 to$10.95 and this means that the entire drop in the price of the 10,000 optionswill be $8900. Taking both of these effects into account, the portfolio willthen be worth $776,956. The overall value of the portfolio will not change (towithin $44 due to round-off effects) regardless of what happens to the stockprice. If the stock price increases, there is an offsetting loss on the option; if

  • 24 CHAPTER 1. BACKGROUND IDEAS

    Original Portfolio S = 100, C = $11.848,944 shares of stock $894,400Short position on 10,000 options -$118,400Net value $776,000

    Stock Price rises 1% S = 101, C = $12.738,944 shares of stock $903,344Short position on 10,000 options -$127,300Net value $776,044

    Stock price falls 1% S = 99, C = $10.958,944 shares of stock $885,456Short position on options -$109,500Net value $775,956

    Table 1.2: Hedging to keep a portfolio constant.

    the stock price falls, there is an offsetting gain on the option.This example is not intended to illustrate a prudent investment strategy.

    If an investor desired to maintain a constant amount of money, putting thesum of money invested in shares into the bank or in Treasury bills insteadwould safeguard the sum and even pay a modest amount of interest. If theinvestor wished to maximize the investment, then investing in stocks solelyand enduring a probable 10% loss in value would still leave a larger totalinvestment.

    This example is a first example of short selling. It is also an illustration ofhow holding an asset and short selling a related asset in carefully calibratedratios can hold a total investment constant. The technique of holding andshort-selling to keep a portfolio constant will later be an important idea inderiving the Black-Scholes formula.

    Sources

    The section on risk and uncertainty is adapted from N. Silver, The Signaland the Noise, The Penguin Press, New York, 2012, page 29. The exam-ples in this section are adapted from Options, Futures and other DerivativeSecurities by J. C. Hull, Prentice-Hall, Englewood Cliffs, New Jersey, 1993and The Mathematics of Financial Derivatives by P. Wilmott, S. Howison,J. Dewynne, Cambridge University Press, 1995, Section 1.4, What are op-tions for?, Page 13, and Financial Derivatives by Robert Kolb, New York

  • 1.3. SPECULATION AND HEDGING 25

    Institute of Finance, New York, 1994, page 110.

    Key Concepts

    1. Options have two primary uses, speculation and hedging.

    2. Options can be a cheap way of exposing a portfolio to a large amountof risk. Sometimes a large amount of risk is desirable. This is the useof options and derivatives for speculation.

    3. Options allow the investor to insure against adverse security valuemovements while still benefiting from favorable movements. This isuse of options for hedging. This insurance comes at the cost of buy-ing the option.

    Vocabulary

    1. Risk is random financial variation that has a known (or assumed)probability distribution. Uncertainty is chance variability that is dueto unknown and unmeasured factors.

    2. Speculation is to assume a financial risk in anticipation of a gain,especially to buy or sell to profit from market fluctuations.

    3. Hedging is to protect oneself financially against loss by a counter-balancing transaction, especially to buy or sell assets as a protectionagainst loss because of price fluctuation.

    Problems to Work for Understanding

    1. You would like to speculate on a rise in the price of a certain stock. Thecurrent stock price is $29 and a 3-month call with strike of $30 costs$2.90. You have $5,800 to invest. Identify two alternate strategies, oneinvolving investment in the stock, and the other involving investmentin the option. What are the potential gains or losses from each due toa rise to $31 in three months? What are the potential gains or lossesfrom each due to a fall to $27 in three months?

  • 26 CHAPTER 1. BACKGROUND IDEAS

    2. A company knows it is to receive a certain amount of foreign currencyin 4 months. What kind of option contract is appropriate for hedging?What is the risk? Be specific.

    3. The current price of a stock is $94 and 3-month call options with astrike price of $95 currently sell for $4.70. An investor who feels thatthe price of the stock will increase is trying to decide between buying100 shares and buying 2,000 call options. Both strategies involve aninvestment of $9,400. Write and solve an inequality to find how high thestock price must rise for the option strategy to be the more profitable.What advice would you give?

    Outside Readings and Links:

    Speculation and Hedging A short youtube video on speculation andhedging, from The Trillion Dollar Bet.

    More Speculation and Hedging A short youtube video on speculationand hedging.

    1.4 Arbitrage

    Section Starter Question

    Its the day of the big game. You know that your rich neighbor really wantsto buy tickets, in fact you know hes willing to pay $50 a ticket. While oncampus, you see a hand lettered sign offering two general-admission ticketsat $25 each, inquire immediately at the mathematics department. You haveyour phone with you, what should you do? Discuss whether this is a frequentevent, and why or why not? Is this market efficient? Is there any risk in thismarket?

    Definition of Arbitrage

    The notion of arbitrage is crucial in the modern theory of finance. It is acornerstone of the Black, Scholes and Merton option pricing theory, devel-oped in 1973, for which Scholes and Merton received the Nobel Prize in 1997(Fisher Black died in 1995).

    http://www.youtube.com/watch?v=G17rx7H3DtIhttp://www.youtube.com/watch?v=xxZtWDJGEvA

  • 1.4. ARBITRAGE 27

    An arbitrage opportunity is a circumstance where the simultaneouspurchase and sale of related securities is guaranteed to produce a risklessprofit. Arbitrage opportunities should be rare, but in a world-wide basissome do occur.

    This section illustrates the concept of arbitrage with simple examples.

    An arbitrage opportunity in exchange rates

    Consider a stock that is traded in both New York and London. Suppose thatthe stock price is $172 in New York and 100 in London at a time whenthe exchange rate is $1.7500 per pound. An arbitrageur in New York couldsimultaneously buy 100 shares of the stock in New York and sell them inLondon to obtain a risk-free profit of

    100 shares 100 /share 1.75 $/ 100 shares 172 $/share = $300

    in the absence of transaction costs. Transaction costs would probably elim-inate the profit on a small transaction like this. However, large investmenthouses face low transaction costs in both the stock market and the foreignexchange market. Trading firms would find this arbitrage opportunity veryattractive and would try to take advantage of it in quantities of many thou-sands of shares.

    The shares in New York are underpriced relative to the shares in Londonwith the exchange rate taken into consideration. However, note that thedemand for the purchase of many shares in New York would soon drivethe price up. The sale of many shares in London would soon drive the pricedown. The market would soon reach a point where the arbitrage opportunitydisappears.

    An arbitrage opportunity in gold contracts

    Suppose that the current market price (called the spot price) of an ounce ofgold is $398 and that an agreement to buy gold in three months time wouldset the price at $390 per ounce (called a forward contract). Suppose thatthe price for borrowing gold (actually the annualized 3-month interest ratefor borrowing gold, called the convenience price) is 10%. Additionallyassume that the annualized interest rate on 3-month deposits (such as acertificate of deposit at a bank) is 4%. This set of economic circumstancescreates an arbitrage opportunity. The arbitrageur can borrow one ounce of

  • 28 CHAPTER 1. BACKGROUND IDEAS

    time3 months

    $398

    $9.95

    $390

    $398 + $3.98

    $2.03

    Figure 1.3: A schematic diagram of the cash flow in the gold arbitrage ex-ample.

    gold, immediately sell the borrowed gold at its current price of $398 (thisis called shorting the gold), lend this money out for three months andsimultaneously enter into the forward contract to buy one ounce of gold at$390 in 3 months. The cost of borrowing the ounce of gold is

    $398 0.10 1/4 = $9.95and the interest on the 3-month deposit amounts to

    $398 0.04 1/4 = $3.98.The investor will therefore have 398.00+3.989.95 = 392.03 in the bank ac-count after 3 months. Purchasing an ounce of gold in 3 months, at the forwardprice of $390 and immediately returning the borrowed gold, he will make aprofit of $2.03. This example ignores transaction costs and assumes interestsare paid at the end of the lending period. Transaction costs would probablyconsume the profits in this one-ounce example. However, large-volume gold-trading arbitrageurs with low transaction costs would take advantage of thisopportunity by purchasing many ounces of gold.

    Figure 1.3 schematically diagrams this transaction. Time is on the hori-zontal axis, and cash flow is vertical, with the arrow up if cash comes in tothe investor, and the arrow down if cash flows out from the investor.

    Discussion about arbitrage

    Arbitrage opportunities as just described cannot last for long. In the firstexample, as arbitrageurs buy the stock in New York, the forces of supply

  • 1.4. ARBITRAGE 29

    and demand will cause the New York dollar price to rise. Similarly as thearbitrageurs sell the stock in London, they drive down the London sterlingprice. The two stock prices will quickly become equal at the current exchangerate. Indeed the existence of profit-hungry arbitrageurs (usually pictured asfrenzied traders carrying on several conversations at once!) makes it unlikelythat a major disparity between the sterling price and the dollar price couldever exist in the first place. In the second